[Disclaimer: I am not here to argue which sect is correct (although you can clearly see Vaishnava bias in many of the sources I present). I am just using my knowledge of scripture to defend all worshipers alike.]
First, it is important to understand the different objects these epithets refer to. Let's take the name Indra for example. Indra is both a god (the king of heaven) and a name denoting Brahman. When Indra is portrayed as a womanizer, that refers to the king of heaven, not to the spotless Brahman in the Vedas. We will look to the Brahma Sutras for further elaboration (since the existence of Brahman has been assumed in the question).
In the Govinda-bhāṣya (with expanded commentary), there is the following on Adhyaya 2 Pada 3 Sutra 15:
Viṣaya [thesis or statement]: The Holy Names of the Supreme Personality of Godhead are eternal and innumerable. During the
temporary manifestation of the material creation, some of these names
are also used to refer to material personalities and objects. But the
primary meaning of these words remains the Lord, since at the end of
the creation the material persons and objects cease to exist.
Saṁśaya [arisal of doubt]: Is it not so that if Lord Hari is the
Supreme Personality of Godhead, the master of all, and the
all-pervading Supersoul, then the names of all that is moving and
inert would also be names of Him?
Pūrvapakṣa [antithesis]: It is not true that all names are names of
the Lord, for words are primarily the names of the various moving and
inert things. We accept the primary meaning of words as given in the
dictionary, and if they also sometimes indicate Lord Hari, that is a
secondary or indirect meaning.
Siddhānta [Vedic conclusion]: Thinking that someone may accept this
idea that words are primarily names of various things and only
secondarily names of the Supreme Personality of Godhead, the author of
the sūtras gives the following explanation.
Indeed, He resides in all that move and does not move. Therefore it will be learned that every word is one of His names.
...The sūtra explains: bhāva-bhāvitvāt
[the real meaning of names will be learned in the future]. This means that by studying the scriptures one will
come to understand that all words are names of the Supreme Personality of Godhead.
From this, it is clear that every word primarily denotes Brahman and secondarily the other bearer of the name. Indra can denote both a deity and Brahman. Demeaning the deity is not equivalent to reducing the purity of the name itself.
At this point you may object, "You have still failed to justify the deification of epithets, which is prevalent throughout the Puranas." I take it that you believe the conception of gods/deities is unfounded in scripture, and only Brahman is described. Then I can justify my position in two ways.
(1) Citing the authority of the scriptures in question:
One should clarify the meaning of the Vedas by the Itihasas and Puranas.
Brihad-aranyaka Upanishad 2.4.10:
The breathing of the Lord is the Rig, Yajur, Sama, and Atharva Vedas, the Itihasas and Puranas.
Note that the Itihasas, which also include narrations of deities, are mentioned as well.
Skanda Purana 2.3-5:
Long ago the grandfather of the devatas performed intense austerities. The Vedas with their six angas appeared with their verses in particular order. Then all the eternal, pure Puranas, embodiment of all scriptures, with eternal words, of a billion verses, emerged from Brahma's mouth. Please hear about the different Puranas starting with the Brahma Purana.
There are two things especially worth noting here. 1) The word "devatas" is mentioned, affirming the existence of deities such as Brahma and Indra. 2) The Puranas are stated to have come from Brahma's mouth. The Puranas are not of human origin, just like the Vedas themselves.
Shrimad Bhagavatam 3.12.39:
Brahma, who has faces in all directions, created the fifth Veda composed of the Puranas and Itihasas, from all of his mouths.
Chandogya Upanishad 7.1.2:
I studied the Rig Veda, Yajur Veda, Sama Veda, Atharva Veda, the Itihasas and Puranas, the fifth of Veda.
Matsya Purana 53.8-9:
Thinking that in the passing of time people cannot receive the Puranas, I [Vishnu] appear in the form of Vyasa and collate them yuga after yuga.
Shiva Purana 1.33-34:
The Lord, condensing the four Vedas, divided them into four. He is known as Veda-vyasa because he divided (vyasta) the Vedas. Being only four hundred thousand verses, the Puranas were condensed by him. Even today in heavenly planets there are a billion verses.
I have obtained all these verses from the Tattva Sandarbha of Jiva Gosvami.
As you can see, a wide variety of texts prove that the Puranas (and Itihasas) are self-evident scriptures like the four Vedas. Both Brahma and the wise rishis manifested the eternal Vedas; likewise, both Brahma and Vyasa manifested the eternal Puranas.
When both the Puranas and the Itihasas "deify epithets" and are shown to be the fifth Veda, there is no question of either their authenticity or the existence of devatas.
(2) Reinterpreting the statement in the Rig Veda:
You cited Rig Veda I.164.46 in your question:
एकं सद विप्रा बहुधा वदन्त्यग्निं
ekaṃ sad viprā bahudhā vadantyaghniṃ
Though it is One, inspired poets speak of it in many ways.
Now that the authority of the Puranas has been established, let us "clarify the meaning of the Vedas" with two Puranic statements.
Viṣṇu Purāṇa [1.2.3] (This is taken from the Govinda-bhāṣya mentioned earlier):
The Supreme Personality of Godhead is one, although He has many forms.
Shrimad Bhagavatam 3.32.36:
The Supreme Personality of Godhead alone is complete transcendental knowledge, but according to the different processes of understanding He appears differently, either as impersonal Brahman, as Paramātmā, as the Supreme Personality of Godhead or as the puruṣa-avatāra.
The gist of it is this: let us assume that Vishnu is verily Brahman for a second (since these Puranas are eulogizing Vishnu). If He is the One mentioned in your Rig Veda quote, then it is very simple to reconcile the fact that Brahman is spoken of in many ways, for Vishnu has many incarnations spoken of in the Puranas. Shaivas can also claim the same about Shiva, and the same goes for other sects.
In conclusion, the deification of epithets is justified in the Puranas because the deities do exist, but it is not "deifying an epithet" per se. The names originally belong to Brahman and are given to other objects when they come into existence in the cosmic manifestation. Additionally, since the Puranas are of the same nature as the Vedas, they can be used to understand the latter's purport. Hence, followers of different sects are not anti-Hindu.
EDIT: Neo-Advaitins and the like would respond like so: the one Brahman manifests as the gods Indra, Agni, Shiva, Vishnu, etc. to perform lilas (divine play), so when in one purana Shiva is called supreme and in another purana Vishnu is called supreme, there is no contradiction. Indeed, both are manifestations of the same Brahman. This is the explanation I have seen.